element_orders

std.algebra.element_orders · Level L3

The order of every element of a finite group: the least k ≥ 1 with gᵏ = e, found by checking every smaller power. A Scan of power_step over k = 1 … n; in a group every order is at most n and divides n (0 would mean no power returned to e, which a group rules out).

ord(g) = min{k ≥ 1 : gᵏ = e}

Signature

element_orders(T: i64[n, n]) → f64[n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference repeated multiplication by g until e, in Python in exact rational arithmetic, on all 40 test cases.
All 159 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

group_axioms power_step

Called by

element_of_order

sha256:f3501344967774ac44e0c3c0e00889c23eeb4c760593a53ae7bb5defe2e015cb

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.element_orders · Ω:element_orders
Pins: sha256:6ffa5a98cccc53135f4cce44ad79d89b1a733fcb214a7bd9ec309b3fbed94bbbthis graph and the 2 it reaches through calls
Evidence: sha256:83ace56a6f9f48d34900aa295ef965417d6e9be93a734075ee0736730eb9d211the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.